On Limitations to the Achievable Path Following Performance for Linear Multivariable Plants

In this paper, we consider a problem termed ldquopath followingrdquo. This differs from the common problem of reference tracking, in that here we can adjust the speed at which we traverse the reference trajectory. We are interested in ascertaining the degree to which we can track a given trajectory, and in characterizing the class of paths for which we can generate an appropriate temporal specification so that the path can be tracked arbitrarily well in an
L

₂
sense. We give various bounds on the achievable performance, as well as tight results in special cases. In addition, we give a numerical procedure based on convex optimization for computing the achievable performance. The results demonstrate that there are situations where arbitrarily good
L

₂
performance may be achieved even though the origin is not in the convex hull of the positive limit set of the path to be followed.